SOLUTION: if 2log5(2x-y)=log5 xy^2 - log5 y , find x/y (5 = base)

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Question 1080330: if 2log5(2x-y)=log5 xy^2 - log5 y , find x/y
(5 = base)

Answer by Edwin McCravy(20062) About Me  (Show Source):
You can put this solution on YOUR website!
 2log%285%2C%282x-y%29%29%22%22=%22%22log%285%2C%28xy%5E2%29%29+-+log%285%2C%28y%29%29

 log%285%2C%282x-y%29%5E2%29%22%22=%22%22log%285%2C%28%28xy%5E2%29%2Fy%5E%22%22%29%29

 %282x-y%29%5E2%22%22=%22%22%28xy%5E2%29%2Fy%5E%22%22

 %282x-y%29%5E2%22%22=%22%22%28xy%5Ecross%282%29%29%2Fcross%28y%5E%22%22%29

 %282x-y%29%5E2%22%22=%22%22xy

 4x%5E2-4xy%2By%5E2%22%22=%22%22xy

 4x%5E2-5xy%2By%5E2%22%22=%22%220

 4x-y%29%28x-y%29%22%22=%22%220



However, such equations often have extraneous 
answers, so we must check.  Logarithms may
be taken only of positive numbers.

Checking
x%2Fy%22%22=%22%221%2F4
y%22%22=%22%224x
2log%285%2C%282x-y%29%29%22%22=%22%22log%285%2C%28xy%5E2%29%29+-+log%285%2C%28y%29%29
2log%285%2C%282x-4x%29%29%22%22=%22%22log%285%2C%28x%284x%29%5E2%29%29+-+log%285%2C%284x%29%29
2log%285%2C%28-2x%29%29%22%22=%22%22log%285%2C%28x%2816x%5E2%29%5E%22%22%29%29+-+log%285%2C%284x%29%29
2log%285%2C%28-2x%29%29%22%22=%22%22log%285%2C%2816x%5E3%29%29+-+log%285%2C%284x%29%29
We can see that on the left side, x must be negative 
to insure that a logarithm is being taken only of a 
positive number.  However that causes both terms on 
the right to be logarithms of negative numbers. 

So we must discard x%2Fy%22%22=%22%221%2F4

Checking 
x%2Fy%22%22=%22%221
y%22%22=%22%22x
2log%285%2C%282x-y%29%29%22%22=%22%22log%285%2C%28xy%5E2%29%29+-+log%285%2C%28y%29%29
2log%285%2C%282x-x%29%29%22%22=%22%22log%285%2C%28x%28x%29%5E2%29%29+-+log%285%2C%28x%29%29
2log%285%2C%28x%29%29%22%22=%22%22log%285%2C%28x%5E3%29%29+-+log%285%2C%28x%29%29
2log%285%2C%28x%29%29%22%22=%22%22log%285%2C%28x%5E3%2Fx%29%29
2log%285%2C%28x%29%29%22%22=%22%22log%285%2C%28x%5E2%29%29
2log%285%2C%28x%29%29%22%22=%22%222log%285%2C%28x%29%29

Which checks.

The only solution is x%2Fy%22%22=%22%221

Edwin